You can answer by filling in the blank spaces. If there is not enough
space attach other sheets.

In this lab we will work on two
things: weighted voting systems, and fair division. This first section
will deal with weighted voting systems.

In this part of the lab, you will
explore how much power different voters hold
in systems where they have an unequal number of votes, such as
shareholders' meetings, or the Electoral College, or many kinds of
boards.

In this lab (and in this problem
set), we just count, for each voter, the number of coalitions where
that voter holds a swing vote, and we do not divide by anything. In any
case, it is only the relative power of all the different players that
really matters; voter A has twice as much power as B if the number of
coalitions in which A has a swing vote is double the number of
coalitions in which B has a swing vote. You can check this ratio (2 if
A has twice as much power as B) by comparing either the straight
numbers of coalitions, or also by using the "normalized" ratios above
(the denominators cancel out).

in Banzhaf's original definition
(also the one used in "Rational Politics" by Steven Brams), yet a
different denominator is used. There one computes the sum N, over all
the voters, of the number of coalitions where that voter has a swing
vote, and uses this for the denominator; the index is then given by

In this case, the sum of the power
indices of all the voters is always 1.

In the first example on the web it is
explained how to compute the relative powers of a system with 3 voters,
one of which holds two votes, while
the other two have both just one. A measure can pass only if at least 3
votes are cast in favor of it.

This system is denoted

[3:2,1,1]

the first 3 denotes the majority
necessary; the numbers after the colon give the number of votes held by
the different players.

After this is explained in detail,
you will be asked to figure out for yourself the indices for other
systems. In this problem set, you will calculate the indices for
different systems.

Problem 1. Calculating Banzhaf
Power Indices.

The first one is [51: 48,45,7] : a
system with three voters
again, with relative weights 48, 45 and 7, and where a 51
majority is needed to pass a measure.

To find the power indices, we start
by making a table as in the [3:2,1,1] example:

Complete table

A

B

C

Number of Votes

Pass/Fail

48

45

7

y

y

y

100

pass

y(s)

y(s)

n

93

pass

y(s)

n

y(s)

55

pass

n

y

y

y

n

n

n

y

n

n

n

y

n

n

n

Again, you compute on every line how
many votes have been
cast in favor of the measure (yes votes), and whether the
measure passes or fails (that is, whether it got 51 votes or more).
Then you check for each voter whether the status
(pass or fail) of the measure would change if that voter
had a mind change. You could put a little (s) next to each swing vote
(see the first three lines in the example)
to make sure you don't forget any, but this is optional.

After you have done all the lines,
count the number of
(s) marks for each voter.

What is your final answer?

Answer:

[51: 48,45,7] =

Problem 2. Same example with a new
quorum.

The next example is [60: 48,45,7] .
The three voters have the same weights as in the previous question, but
the majority rule has changed. Now 60 votes are necessary to pass a
measure.

Complete the following table again:

A

B

C

Number of Votes

Pass/Fail

48

45

7

y

y

y

y

y

n

y

n

y

n

y

y

y

n

n

n

y

n

n

n

y

n

n

n

And give here your final answer:

Answer:

[60:48,45,7] =

Problem 3. Another example of
calculating BPI's.

Next: [51: 35,35,30]

Complete the following table again:

A

B

C

Number of Votes

Pass/Fail

35

35

30

y

y

y

y

y

n

y

n

y

n

y

y

y

n

n

n

y

n

n

n

y

n

n

n

And give here your final answer:

Answer:

[51: 35,35,30] =

Compare this with the first example,
[51: 48,45,7] that you worked out?
In particular, compare the amount of power each voter has versus the
percentage of the votes they have in the two cases.

Answer:

Problem 4. An example with four
voters.

This example has four voters: [60:
31,31,31,28].The table has now 16 lines:

Complete the table

A

B

C

D

Number of Votes

Pass/Fail

31

31

31

28

y

y

y

y

y

y

y

n

y

y

n

y

y

n

y

y

n

y

y

y

y

y

n

n

y

n

n

y

n

n

y

y

n

y

n

y

y

n

y

n

n

y

y

n

y

n

n

n

n

y

n

n

n

n

y

n

n

n

n

y

n

n

n

n

What is your answer now?

Answer:

[60: 31,31,31,28] =

Any comments? In particular, is the
distribution of power what you would
expect given the percentages of the votes each voter has?

Answer:

Problem 5. Maximum BPI.

Assuming that the quorum is greater
than or equal to the simple majority, if one of the voters has the
maximum Banzhaf Power Index (as determined by the number of voters),
explain why the other voters have zero power.

Answer:

Next you can experiment with some
more complicated systems, with up to 10 voters on the webpage Create Your
Own Example of Weighted Voting System. You should enter the
different weights yourself in the lab. If there are fewer than 10
voters, enter the weights (from left to right) in the boxes and leave
the unused boxes
empty (except the first box, which must be filled in) - the system will
ignore
the empty boxes (as it will voters with zero weight). The "Banzhaf
power index" (BPI) calculated by the software is the total number of
coalitions that
a voter can swing.

If you are interested in simple
majority rule, then you can click the button "Simple majority", and the
required majority quota will appear; if you want to
work with another quota, then you have to enter it by hand. When you
then click
"Power Index", you will see the answer appear. To work with a slightly
changed
example, you don't need to click "Clear Table" and enter everything
again; you
can just go back to the boxes with entries that will change, and change
them
there; if you then click "Power Index" again, you get the new result.

Problem 6. Blockvotia Example.

In the Scientific American article
that was handed out in class, different weights were given to six
districts in
fictional Blockvotia.

a) At first, the distribution
of weights was

Sheepshire

10

Richfolk

9

Candlewick

7

Fiddlesex

3

Slurrey

1

Porkney Isles

1

What are the power indices of the six
districts under
simple majority rule?

Sheepshire

Richfolk

Candlewick

Fiddlesex

Slurrey

Porkney Isles

b) The two smallest districts
complained, and lobbied hard
to get each an extra vote, so that the distribution of weights would be

Sheepshire

10

Richfolk

9

Candlewick

7

Fiddlesex

3

Slurrey

2

Porkney Isles

2

What would the power indices now be
under simple majority
rule?

Sheepshire

Richfolk

Candlewick

Fiddlesex

Slurrey

Porkney Isles

c) But this solution turned out
to be infeasible. However,
if the biggest district Sheepshire is given two extra votes,
so that the distribution is

Sheepshire

12

Richfolk

9

Candlewick

7

Fiddlesex

3

Slurrey

1

Porkney Isles

1

then the following power indices are
obtained (again under
simple majority rule):

Sheepshire

Richfolk

Candlewick

Fiddlesex

Slurrey

Porkney Isles

Can you compare this with the previous
situations?

Answer:

Problem 7. Power in the European
Community.

Next, let us experiment with the
European Community numbers
illustrated in the class handout.

a) Originally there were 6
countries:

Weight

France

4

Germany

4

Italy

4

Belgium

2

Netherlands

2

Luxembourg

1

The quorum needed to pass a measure
was 12 (out of 17, which is much larger than simple majority!). What
are the corresponding power indices?

Answer:

b) Experiment with raising and
lowering the quorum. How does that affect Luxembourg's power? And that
of the other countries? (Give the numerical results!)

Answer:

c) Next, three new members
were added, England, Denmark and
Ireland, with a redistribution of the weights, so that
we get

Weight

France

10

Germany

10

Italy

10

Belgium

5

Netherlands

5

Luxembourg

2

England

10

Denmark

3

Ireland

3

and the new quorum is 41. How does
the power of Luxembourg fare now? Compare it with that of the other
countries.

Answer:

d) Later, Greece was added,
and the new list (with weights) was

Weight

France

10

Germany

10

Italy

10

Belgium

5

Netherlands

5

Luxembourg

2

England

10

Denmark

3

Ireland

3

Greece

5

with quorum 45. What is the power
distribution now? Compare Luxembourg's power with that of other
countries.

Answer:

This second part of the lab deals
with Fair Division. For this part of the lab, you won't need anything
from the web.

Problem 8. Fair 4-way division of
an inheritance.

We are going to work through an
example of fair division in an inheritance, involving four people,
Janice, Scott, Andrea and Eric.

The fair division procedure concerns
three big items, the house in the small town in which they all grew up,
a very nice cabin in the mountains where they spent many vacations when
they were children, and a boat in which they all remember going on
fishing trips.

Each of the four is asked to assign
their own subjective value to the house. All of the different items
need some work, and they are probably worth more to
them, effectively, than the market value if they just simply got them
appraised.

They all have different preferences
as well; Eric has remained in the same area of the country and he has a
lot of affection for the house in which he grew up. His own family is
growing, and they need more space. He really would
like to move into his parents' house. Janice doesn't care much about
the house
- she certainly doesn't want to live in it, and she knows it would need
a lot of fixing up before they could get a good price for it. On the
other hand, she
has fond memories of the mountain cabin, and if she could get it, she
would use it often for hiking get-aways from her city job. Scott
wouldn't mind taking over the house, but he does not feel about it very
strongly; he certainly doesn't intend to take vacations in the mountain
cabin - his job
doesn't leave him enough time to go there often for a few days, and he
prefers
to spend longer vacations elsewhere. But he is very much interested in
the boat, with which he could go on day-outings during the weekend.
Finally, Andrea
is not so determined to take over the boat or the house or the cabin,
and she just tries to gauge what she thinks would be a reasonable price
for them.

When they are asked to put a money
value on each item, taking into account not only what they think it is
worth, but also what it is worth to THEM, they come up with:

house

cabin

boat

Eric

200,000

40,000

16,000

Janice

100,000

80,000

20,000

Scott

180,000

60,000

32,000

Andrea

140,000

60,000

24,000

They have to make these evaluations
separately and simultaneously - no jockeying for position or lying
about
one's own interest after having had a peek at what the others
said.

a) Try now to divide up this
estate according to the Knaster procedure that we saw in class.

The main steps are :

(i) assign each item to the person
who wanted it most

Answer:

who gets the house?

who gets the cabin?

who gets the boat?

(ii) imagine that each of them now
pays into a common "pot" the amount corresponding to what they got
"over" their share. For instance, Eric gets the house, worth in his
estimate 200,000, of which his share would normally have been 25%; so
he has to pay the excess, 75% of 200,000, into the pot.

Answer:

amount Eric pays into the pot:

amount Janice pays into the pot:

amount Scott pays into the pot:

amount Andrea pays into the pot:

(iii) next, each person withdraws
from the pot the money equivalent, according to their own evaluation,
of the shares in the items that they didn't get.

Answer:

amount Eric withdraws from pot at
this stage:

amount Janice withdraws from pot
at this stage:

amount Scott withdraws from pot
at this stage:

amount Andrea withdraws from pot
at this stage:

(iv) what remains in the pot after
this stage gets divided up evenly between all the heirs.

amount each of them gets at this
step:

Answer:

Eric:

Janice:

Scott:

Andrea:

(v) compute for everyone what they
got, in items (if they got any), and in money (include in this the
monetary value of any item a person got. At least one of them will have
to pay the others money. You can use negative amounts to represent
money owed or paid.):

Answer:

Eric:

Janice:

Scott:

Andrea:

For each of them, compare the value
of what they got (according to their own evaluation again) with what
they thought the whole estate was worth.

Answer:

Eric:

Janice:

Scott:

Andrea:

b) Imagine now a slightly
different situation. We still have the same three items, and the same
evaluations by each of the four heirs. But, because Eric spent much
more time in the final years of their parents' life taking care of many
things for them, all the children have agreed that this entitles him to
40% of the estate, while the other three get 20% each. Please work
through the whole division again with these unequal weights.

Answer:

c) If you have any comments
about the whole procedure, please give them here. Include in your
comments whether or not you think this scheme is envy-free?

Answer:

Challenge Questions

Problem C1. Combinations of BPI's.

For three voters, give all possible
combinations of Banzhaf power indices that
can occur. Explain first why all the power indices are even numbers.

Answer:

Problem C2. Strategic evaluations
in "fair" division.

In Problem 8. a) above, on the fair
division of the inherited estate equally among the four siblings,
consider what Eric could do to maximise his gain if he had secretly
seen the evaluations of each of his other siblings before making his
own evaluations.